A projective point is a line in ir3 that passes through the origin. We then formalize pappus property as well as hexamys in order to prove hessenberg theorem, which states that pappus property entails desargues property in projective plane. Theorem 2 is false for g 1 since in that case t p2gk is a discrete poset. Hence a classical projective plane can be defined as such a. Any two points p, q lie on exactly one line, denoted pq.
Given any figures drawn on a flat plane surface s, we can imagine this plane embedded in threedimensional space, and we can select an arbitrary point p in the space not on s and some other flat plane surface s. Proving and generalizing desargues twotriangle theorem. We give a short proof to pappuss hexagon theorem one of the most famous and ancient results of projective geometry. Projective geometry in a plane fundamental concepts undefined concepts. Noneuclidean geometry the projective plane is a noneuclidean geometry. Theorems of desargues and pappus the elegance of these statements testifies to the unifying power of projective geometry. These two approaches are carried along independently, until the. Lesson plans for projective geometry 11th grade main lesson last updated november 2016 overview in many ways projective geometry a subject which is unique to the waldorf math curriculum is the climax of the students multiyear study of geometry in a waldorf school. No distances, no angles, no right angles, no parallel lines. What other properties are preserved under the allowed transformations. If the image of g is not contained in a line, then there exists a semilinear map f. Theorem 2 fundamental theorem of symplectic projective geometry. Original proof of pappus hexagon theorem mathoverflow. There appear in the literature one or two references to dates for pappus s life which must be wrong.
The elegance of their proofs testifies to the power of the method of homogeneous coordinates. Nothing is known of his life, except from his own writings that he had a son named hermodorus, and was a teacher in alexandria. The theorem that we will investigate here is known as pappuss hexagon. Pascals result is proved using projective methods, in particular, using desargues idea of. Pappus of alexandria is the last of the great greek geometers and one of his theorems is cited as the basis of modern projective geometry. Drawing of the theorem of pappus starting with a at infinity. A first look at projective geometry, starting with pappus theorem, desargues theorem and a fundamental relation between quadrangles and quadrilaterals. Section 6 and is the key to our proof of pappus theorem. Theorem 2 is a consequence of the fundamental theorem of projective geometry see. Desargues theorem, pappuss theorem, pascals theorem, brianchons theorem 1.
Pv\\e pw a morphism between the associated projective spaces. Journal for geometry and graphics volume 11 2007, no. The pappus geometry configuration has 9 points and 9 lines. Projective geometry, branch of mathematics that deals with the relationships between geometric figures and the images, or mappings, that result from projecting them onto another surface. The following theorem is of fundamental importance for projective geometry.
According to coxeter probability cheat sheet pdf c1, hilbert observed that. The theorem of pascal concerning a hexagon inscribed in a conic. If one restricts the projective plane such that the pappus line is the line at infinity, one gets the affine version of pappus s theorem shown in the second diagram. In mathematics, pappuss centroid theorem also known as the guldinus theorem, pappus guldinus theorem or pappuss theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.
Coxeter provides good coverage of the fundamental concepts of synthetic projective geometry. Girard desargues 1591 1661 father of projective geometry 6. Pappuss projective theorem pappus of alexandria fl. Dorrie begins by providing the reader with a short exposition of. The usual method of proving pappus theorem today is in the context of projective geometry. Chasles et m obius study the most general grenoble universities 3. Theorem asserts that the points a, b, c lie on a straight line. Does anyone know where i can find an english translation, preferably online or in a book the library of a small liberal arts college would be likely to have, of the original proof of pappus hexagon theorem from projective geometry. An analytic proof of the theorems of pappus and desargues. The line lthrough a0perpendicular to oais called the polar of awith respect to. Imo training 2010 projective geometry alexander remorov poles and polars given a circle.
The basic intuitions are that projective space has more points than euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points. Pappuss theorem is a special case of pascals theorem for a conicthe limiting case when the conic degenerates into 2 straight lines. The pascal theorem, a generalization of the pappus. Projective geometry and pappus theorem kelly mckinnie history pappus theorem geometries picturing the projective plane lines in projective geometry back to pappus theorem proof of pappus theorem. If the vertices of a triangle are projected onto a given line, the per pendiculars from the projections to the corresponding sidelines of the triangle intersect at one point, the orthopole of the line with respect to the triangle. An application of pappus involution theorem in euclidean. This book was created by students at westminster college in salt lake city, ut, for the may term 2014 course projective geometry math 300cc01. His great work a mathematical collection is an important source of information about ancient greek mathematics.
The axiomatic destiny of the theorems of pappus and desargues. Theorem 1 fundamental theorem of projective geometry. Pappus theorem the cross ratio of four lines of a pencil of lines equals the cross ratio. Projective geometry is as much a part of a general educa. In class we proved, not exactly their equivalence with thaless theorem, but simply their truth in the geometry of book i. Pappus theorem the theorem has only to do with points lying on lines. Rpn rpn which maps any projective line to a projective line, must be a projective linear transformation. The theorems are attributed to pappus of alexandria and paul guldin. Theorem 2 pappus involution theorem the three pairs of oppo site sides of a complete quadrangle meet any line not through a vertex in three pairs of an involution. Any two lines l, m intersect in at least one point, denoted lm.
Such a set of axioms was given by bachmann, and a proof of desargues theorem can be found in based on the following pappus theorem of euclidean geometry being considered as an axiom. Dec 05, 2008 a first look at projective geometry, starting with pappus theorem, desargues theorem and a fundamental relation between quadrangles and quadrilaterals. The concept of projectivity lies at the very heart. Drawing of the theorem of pappus starting with a and b at infinity. Ai and lines bj in the projective plane, such that, for some pairs of indices. In modern axiomatic projective plane geometry, the theo rems of. Coxeters book, projective geometry second edition is one of the classic texts in the field. Drawing the additional lines shown in figure 1, one sees that the 3 middle. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. A geometry which begins with the ordinary points and lines of euclidean plane geometry, and adds an ideal line consisting of ideal points which are considered the intersections of parallel lines. Thus each equivalence class of parallel lines contains one of these ideal points, which is defined in projective geometry as the intersection of these parallel lines. The pappus pascal proposition is usually called pappus theorem. Note that lines ab and ac must be parallel to line l. Common examples of projections are the shadows cast by opaque objects and motion pictures displayed on a screen.
This geogebrabook contains dynamic illustrations for figures, theorems, some of the exercises, and other explanations from the text. Pappus theorem and the cross ratio universal hyperbolic. Drawing of the theorem of pappus with all three intersecting points at infinity in different directions. In modern axiomatic projective plane geometry, the theorems of pappus and desargues are not equivalent. A case study in formalizing projective geometry in coq. We prove several theorems on orthopoles using the pappus theorem, a fundamental result of projective geometry.
Pdf orthopoles and the pappus theorem semantic scholar. The basic intuitions are that projective space has more points than euclidean space. One can think of all the results we discuss as statements about lines and points in the ordinary euclidean plane, but setting the theorems in the projective plane enhances them. These statements have been generalized and strengthened in numerous ways, and.
We could imagine a geometry in which this does not apply, and in such a context pappus theorem would not be valid. For projective plane geometry, we use a traditional approach dealing with points, lines and an incidence relation to formally prove the independence of desargues property. The difficulty lies in the fact that the homomorphism of division rings associated to the map f can be nonsurjective. The pappus configuration is the configuration of 9 lines and 9 points that occurs in pappuss. The dual of this latter characterization permits to state the projective version of menelaus theorem. Here are some other wellknown theorems from projective geometry. In the axiomatic development of projective geometry, desargues theorem is often taken as an axiom. Apr 20, 2011 pappus theorem is the first and foremost result in projective geometry. First of all, projective geometry is a jewel of mathematics, one of the out standing achievements of. This is a theorem in projective geometry, more specifically in the augmented or extended euclidean plane. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul. Later we shall discuss how the study of projective geometry related to many other subjects including ancient greek mathematics e. Theorem 2 pappus involution theorem the three pairs of opposite sides of a complete quadrangle meet any line not through a vertex in three pairs of an involution.
About 0 years after pappus wrote the collection, an interesting generalization of pappus theorem was discovered by blaise pascal based on the ideas of girard desargues. If points a,b and c are on one line and a, b and c are on another line then the points of intersection of the lines ac and ca, ab and ba, and bc and cb lie on a common line called the pappus line of the configuration. Pappus s projective theorempappus of alexandria fl. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The following version of the fundamental theorem is proved. Projective geometry and pappus theorem kelly mckinnie history pappus theorem geometries picturing the projective plane lines in projective geometry back to pappus theorem proof of pappus theorem pappus of alexandria pappus of alexandria was a greek mathematician. Nov 29, 20 the pappus geometry configuration has 9 points and 9 lines. Pascals theorem is in turn a special case of the cayleybacharach theorem. Another of his significant contributions was the notion of cross ratio of four points on a line, or of four lines through a. In mathematics, pappuss centroid theorem also known as the guldinus theorem, pappusguldinus theorem or pappuss theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.
Theorem 1 the theorem of pappus let be a hexagon with six distinct vertices such that points, and. One starts with 6 points, 3 of which are contained on one line and 3 of which are contained on another. A quadrangle is a set of four points, no three of which are collinear. Topics such as desargues theorem, pappus s theorem and conics are covered. The axiomatic destiny of the theorems of pappus and. You can draw it with a straightedge with no compass. An application of pappus involution theorem in euclidean and. In the western world one usually says pappian geometry, plane, etc. Coxeter projective geometry second edition geogebra. This theorem is an important milestone toward obtaining the arithmetization of geometry which will allow us to provide a connection between analytic and synthetic geometry. If the pappus line and the lines, have a point in common, one gets the socalled little version of pappus s theorem. Pascals result is proved using projective methods, in particular, using desargues idea of points at. Theorem 2 is a consequence of the fundamental theorem of projective geometry see section 6 and is the key to our proof of pappus theorem.
Desarguesian plane which the pappuss theorem is valid. A simple proof for the theorems of pascal and pappus. It is the study of geometric properties that are invariant with respect to projective transformations. On the other hand we have the real projective plane as a model, and use methods of euclidean geometry or analytic geometry to see what is true in that case. This is an important theorem of projective geometry. An elementary proof of the fundamental theorem of projective.
Later, this theorem will play a central role in modern projective geometry. The solutions to some exercises can be found in the back of the book. A course in projective geometry matematik bolumu mimar sinan. Pappus theorem, indicates collinearity of the three intersection points. In 1640, blaise pascal, in his work essays pour les coniques, obtained a result about a hexagon inscribed in a conic that generalizes pappus theorem. Parallelism, which plays a leading role in euclidean plane geometry.
An almost parallel bundle of lines which meets at a point far on the right. A course in projective geometry matematik bolumu, mimar. That is a central topic in projective geometry, and in fact, of any type of geometry. Desargues theorem 1 two triangles said to be perspective from a point if three lines joining vertices of the triangles meet at a corresponding common point called the center or polar point. In class we proved, not exactly their equivalence with thaless theorem, but simply their truth in the geometry of book i of euclids elements.
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